ELI5, How to approach addition and subtraction for a five year old

I was helping a student with math the other day. He’s 5.5 years old and was learning to count and some basics of addition and subtraction.

Counting by 1’s and 2’sHad seen that acronym ELI5 before, so this is literally that.

So he did not have too much of a foundation with substantial repetition or memorization.

However, he can count to 120 fairly accurately and could add numbers with mistakes here and there.

His situation made me think a bit about how counting comes up in different forms in many areas of math. “Fancy counting” was the phrase that Vi Hart used, https://youtu.be/N-7tcTIrers

I was thinking that he was doing pretty well with counting, So that seemed like a good starting place to base addition and subtraction.

What he did with his fingers was interesting to watch. He would use his fingers and then count them all together. So for 5 +3, five fingers on one hand and 3 on the other.

Stemming from that method, he could have the two numbers on each hand. 5 and 3 for example. But then start with the bigger number, beginning with 5 and then counting 6 7 8 rather than starting with 1 and counting all the fingers.

Could do something similar for subtraction. Getting more repetition with counting up and down and starting in different places could also help I think.

“What is the square root of 28 over square root of 7?”

square root 28 over square root 7

Here’s one way to do it, factor the square root of 28, completely and then go on from there.

Another method is a bit less work,

You could place both under the same radical sign, simplify, then solve

square root 28 over square root 7 faster

“What is the order of operations?” PEMDAS, BODMAS

What is the Order of Operations?

It’s basically a system we made up. There isn’t anything wrong with it. But it is just a system for everyone to do things in a standard order. If the system was different, we would simply notate things differently and still be able to do mathematics.

Since there is a standard system, when you see a problem, it’s understood that it should be solved in a certain manner.

BODMAS/PEMDAS

Honestly, I had never seen “BODMAS” before today. But it seems like a valid way to think about the order of operations.

In school I learned PEMDAS. For reference, I was in fifth grade in California in the 1990s.

  1. Parentheses
  2. Exponents
  3. Multiplication & Division
  4. Addition & Subtraction

There are six letters, but I only wrote four numbers with two items on numbers 3 & 4.

The reason for that is that you can think about multiplication and division as essentially being variations of the same operation. Five multiplied by three is the same as five divided by 1/3, etc. And you can either subtract or add a negative number to get the same effect.

Compared to BODMAS

Brackets, Of, Division, Multiplication, Addition, Subtraction

  1. Brackets
  2. Of
  3. Division & Multiplication
  4. Addition & Subtraction

Again, I combined items for #3 and #4, same items.

That acronym includes the word “of” which does come up sometimes. Where it says ‘brackets’, that’s pretty much the same thing as ‘parentheses’ in PEMDAS. There can be a few variations.

Types of Parentheses/Brackets

( )

[ ]

{ }

For either brackets or parentheses, I would pay more attention to their location. You do the inner sets of parentheses first.

And the notation can become easier to follow if you use more than one type of parentheses/brackets.

An Example,

3-3×6+2

Do the multiplication first and then the addition/subtraction.

3 – 18 + 2

Once you only have addition/subtraction, the order does not matter too much.

Personally, I would add the 3 and the 2 first in my head and then subtract 18. It can be done other orders.

5- 18 = -13

“What is the opposite of natural log?”

e to the x and ln x

If you raise the natural number e (approximately 2.7) to the lnx, you get back x.

And if you take the natural log of e^x, you get x.

Getting to 184,029 Points on Khan Academy

About two years ago, I signed up for the online program Khan Academy, created by Salman Khan.

The program has changed a bit since then, partially due to a sizable amount of donations from both Google and Bill Gates among others.

I have explored the program a bit and found it useful, therefore deciding to recommend it to others. Getting to the number of points that I have took a little time, so I feel like I have a pretty good idea of the interface and how things work inside it.

Last year I had my physics class all sign up for accounts since it seemed like a tool that could be useful for them. With videos on physics as well as exercises to review some mathematics.

More recently I had a student that I tutor in math sign up for an account to get more practice.

Strengths of the program, my thoughts

Khan Academy began with a set of tutorial videos that continues to grow and expanded with a bit of programming that gives game-like elements to the math exercises. (Similar in some ways to Duolingo). More recently they have added videos/exercises for computer programming.

Because of the emphasis on the mathematics and expanse of that section, the mathematics area seems to me to be the most useful part of the site. In a similar way, the programming section allows people to try out coding themselves.

The mathematics section also has a ‘pretest’ to somewhat skip a student past a few things they may have mastered quite some time ago.

And there are ‘mastery challenges’ available that will move you through the sections a bit more quickly if you feel you want to move into harder sections for the mathematics.

(Mastery Challenge example, click to see larger image)

khan_mastery_challenge

The tutorial videos can be helpful as well and span over a much greater range of topics than mathematics though.

Here is my status as of right now:

neal_khan_academy_2013-09-27

Explanation of the dashboard

The character/avatar in the top left is something you get to choose. You have more options if you have more points.

Below the words ‘The World of Math’ it shows the numbers 256 on a gray bar and 150 on a dark blue bar with shades of blue in between.

These numbers represent the mathematical topics you have practiced exercises on (gray) and those you have mastered (dark blue). In between are intermediate levels between ‘practiced’ and ‘mastery’.

To the right, there is a large grid of small squares. Each square represents a skill. It shows the same information as that to the left of it, but in more detail. It also shows what is unpracticed. You can hover over those small squares to see what they represent and click on them to go to the lessons and exercises.

The top left square is the most basic and the bottom right is the most advanced, getting into calculus in the later squares.

Above the grid, there are six ‘badges’ with numbers. The badge on the right is the easiest to attain. And the badge second from the left is the hardest to attain, the ‘black hole badges’. Some of the problems refer to these badges, with the ‘earth badges’, ‘moon badges’, etc.

And to the right of the badges is the total number of points. These increase with everything you do- watching videos, working on exercises, etc.

Will it help you?

The math exercises will help act as a supplement to get more practice, for students K-12 up to the early parts of calculus. Practicing problems is an essential part of learning mathematics.

You get instant feedback about whether you have done the problem correctly and can ask for ‘hints’ which will be provided one at a time.

That doesn’t mean it replaces a teacher/tutor since it is limited in the explanations provided. There is a mechanism for allowing users to ask question and provide feedback which does iteratively improves the site.

Sometimes there are mistakes and/or strange ways of setting up problems. You can provide feedback on these things and the people working at Khan Academy have responded to at least one of my suggestions.

If you are past high school, but want to have a more solid foundation in mathematics, I think the program can also help you. It can be a good review/refresher.

Good for more advanced students?

The videos on different topics may be useful for more advanced students. The mathematics section is not designed with a more advanced student in mind though. The math ‘pretest’ will skip past a few things, but not everything you might wish it to, so therefore a more advanced student doing the math problems may find themselves working on things that are fairly basic, even if mostly sticking to ‘mastery challenges’.

The ‘mastery challenges’ take five problems that range in difficulty. At least one should be quite easy and one should be harder, with a range between them.

The program requires a certain amount of time to pass (something like 16 hours) before continuing work on some sections. That is helpful for the K-12 students since the material is more new for them and it’s better not to rush through things too much.

Computer Programming

I’m starting to get farther into this section and have liked what I have seen so far. You learn ideas of javascript for drawing and animation. Exercises and video lessons are mixed together in a sequence.

Seems like a good introduction to this sort of programming.

 

TED Mathematics Talk

Rethinking how math is taught.

He talks about how problems worth solving take time.

The example he uses is from a physics textbook!